Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16285 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,14,6,13)(3,15,18,9,20,42,19,32)(4,10,28,35,24,37,23,36)(5,11,26,41,27,40,29,39)(16,33,46,30,47,43,21,53)(17,48,52,59,51,44,50,31)(22,34,25,56,54,61,55,38)(45,58,64,62,63,60,49,57);(1,3,8,18,7,20,6,19)(2,9,12,42,14,32,13,15)(4,21,28,16,24,46,23,47)(5,17,26,52,27,51,29,50)(10,33,35,30,37,43,36,53)(11,31,41,48,40,59,39,44)(22,49,25,45,54,64,55,63)(34,60,56,57,61,58,38,62);(1,4,7,24)(2,10,14,37)(3,16,20,47)(5,25,27,55)(6,28,8,23)(9,30,32,53)(11,38,40,56)(12,36,13,35)(15,43,42,33)(17,49,51,64)(18,21,19,46)(22,29,54,26)(31,58,59,60)(34,41,61,39)(44,62,48,57)(45,52,63,50);(1,5)(2,11)(3,17)(4,22)(6,29)(7,27)(8,26)(9,31)(10,34)(12,41)(13,39)(14,40)(15,44)(16,45)(18,52)(19,50)(20,51)(21,49)(23,55)(24,54)(25,28)(30,57)(32,59)(33,60)(35,56)(36,38)(37,61)(42,48)(43,58)(46,64)(47,63)(53,62)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363933);